Machine Learning

Machine learning is the latest fad in science. It is roughly speaking the “automated identification of patterns in data.”(MIT) And is indicative of the movement toward big data in many science fields. When looking at the microscopic levels that we do we can get overwhelmed with tremendous data sets and finding subtlety can get quite difficult. The hope is that machines can help guide us to find patterns in data. It can be used to quickly find outliers in a large set of data, or even find mathematical models for data sets. Machine learning is not the same as artificial intelligence which is important to realize. It is simply using machines to find patterns in data. What we use that data for, and what patterns we look for is an important area of study.

With that said there is still a tremendous amount of things we don’t understand about machine learning, we don’t get exactly what the hidden layers are given outputs and inputs, or how efficient it will be. Questions like how many hidden layers should we use? How many nodes should we use? For mathematical models what type of functions should we use to approximate the data? How complex do we want the models to be? Machine learning is the happy medium between Statistics, mathematics, and computer science.

So you want to start using machine learning to up your STEM game, lucky for you, understanding the math isn’t needed unless you want really complex, well designed, efficient machine learning. For most tasks simple models requiring some basics in computer science and algorithms is all that’s needed. But I’m really interested in the math behind machine learning, because the machine part is only a formality. Understanding the math of learning could be the catalyst to understanding how our brains work, why we have a conscience, etc.

Our brains are in a lot of ways like a machine, we have our sensory inputs, our brain does some magic and voila, you make complex decisions, like, this apple is tasty food, whereas this tire is not (hopefully!)

But I would add that we also have internal processes running, regardless of senses. Is that what makes us human? And will machines ever be able to capture that essence? Machine learning makes me appreciate more how complex us meatbags are, it ties so many great field of math together and also manages to find applications to research, helps solve philosophical inquiries, and shows the power of pattern recognition in computer science.

Music Theory

Patterns are all around us, in the tree leaves, waves on a lake, or stock market trends. And as people we are pretty good at detecting these patterns. But even in music. That might be obvious, there’s the beat, rhythm, choruses and rhyming, but in fact there’s a lot deeper of a pattern going on too complex for our conscious, but not for the subconscious.  Whether it is a few notes strung together in a way that fits with the words or the way the melody shifts or suddenly diverts, and why that change is so important, yes music is an art, and in that there is creativity, and likely no AI could replicate that but music theory has its place, when one quantifies art, they can reach a happy middle ground where their imagination and ingenuity guide their focus, but their understanding of the intricacies of music help to express the fine details. Music theory is quantification of art done correctly, I will never be interested enough to come up with my own song based on feel. But I can appreciate brilliance when I learn the theory behind it. It gives more appreciation and also helps to put you in the mindset to be able to distinguish and appreciate the fine details in other parts of your life. Expression is a vital part of everyday life, but being able to clearly describe those expressions is hard, learning the theory behind subjectivity is in of itself nearly impossible, but it helps to build the skills to be creative to express oneself, or to understand ideas instead of things. Despite the seeming lack of relation to patterns and math, that distinction between ideas and things is the very skill that I’ve learned in mathematics, it’s why studying the methods of discovery can help lead to new discovery. It’s not necessarily about the end result, but how they got there that is important.


(still needs revision)

Gödel’s Incompleteness Theorems

Godel’s Incompleteness Theorems are some of the most important results in logic, that I’ve used in many forms day to day. “any formal system (such as a system of the natural numbers), there are certain true statements about the system which cannot be proven by the system itself.” This, although seemingly technical, means to solve a problem you need to look outside the problem itself. It isn’t necessarily the theorem itself, but the state of mind. We often think that we can find the problems and solutions of a problem on whether something is true or not, without looking past the system itself. It is like determining whether the legal system is just based only on the legal system.  Which leads us to the other Incompleteness Theorem, which basically says you can’t determine if something is consistent based on itself. This comes in handy whenever you are trying to figure out how to handle an issue.

The first step when presented with an issue is to find a good goal, or achievement you want to achieve, even if it’s broad, like, betterment for society, as long as there is an outside goal, you can work backwards to determine if you are considering everything.

The other great use, is that you can never fully figure out the impact of a system by looking at the system itself, consider iphones, while we may enjoy them, is our enjoyment worth the bad conditions of the people working to make them? And what can we do about it? Math is at its heart all about problem solving, and I feel like sometimes it gets undervalued to memorization of what dead people figured out and numbers, but really it is about solving problems given axioms. And this result tells us, we never have it all figured out. (and even if we thought we were at least on the right track, we wouldn’t know it)

Prisoner’s Dilemma

Despite our best efforts, it seems math is always there to thwart our brilliant ideas. Whether it be believing in lucky dice rolls or getting on a streak in blackjack, math is there to remind us, it’s all probability. A poor man once said… ” Always bet it all when you are on a winning streak”.

Despite math and logic, we are always looking for the psychological edge and interrogation is a prime example. Consider a scenario where there are two suspects who are believed to have worked together to commit a crime. A clever cop thinks, I know I’ll reward them for telling the truth. The cop individually goes to each suspect with the following…

If one confesses and the other doesn’t, the one who confesses gets off for free, while the other does 10 years. if both confess you each get 6 years, but if neither confess they each get only a year of jail. What would you do?

Believe it or not, you should confess every time. This is because no matter what the other person does, confessing lowers your jail time, and same with the other suspect. This means they will both get 6 years in jail if both individuals act rationally, and self-interested. Whereas cooperation would get them only 1 year!Image result for prisoner's dilemma

This is the so-called prisoner’s dilemma and despite the example being quite specific, it actually comes up quite a bit. It occurs whenever two parties acting bad, is better than a party acting good, while the other is bad. (think politics). This is an important concept which I try to consider whenever proposing an system for people to use, you need to make sure there is an incentive for cooperation rather than taking advantage of others.

This is the sort of thinking you need to have whenever you are in charge of competition, and recognizing little kinks in your proposal which allow for such stalemates can often make a seemingly good plan fail miserably, all because neither group wants to help the other. In many ways this is similar to the government shutdown, in that both groups don’t want to concede, and thus they both end up hurting each other rather than allowing productivity to happen. This is why I love math, because it shows up when you want it to and can often make things more robust, but at the cost of clarity and understanding unfortunately. That needs to change.

Pigeonhole Principle


Math… equations, formulas, integrals, trigonometry. You’re not alone if some of those words give you cognitive dissonance. Unfortunately, math feels like the study of boredom rather than the study of patterns. That is not what math is. Math has had the terrible rap of being all about computation and memorization of complex formulas for solving problems. But why? I think that a lot of it has to do with the fact that it’s harder to teach somebody an idea than it is to teach somebody a formula to remember. But that’s not math. Anyone can remember a formula, but how many can derive it, or make it more general/better.


What is math? Unfortunately, math is tough to decipher. The reason I like it is not the reason other like it, which is not the reason society likes it. Often times it takes so much math for one to see its benefit that it’s not worth going that far into it past basic arithmetic, and maybe trigonometry. I want to build a different perception. I’m gonna talk about a part of math I really enjoy and tell you how it relates to everyday life, I hope it doesn’t get too boring, but it’s math so no promises!


There are two people, (non-bald)in New York with the same number of hairs on their head.

How would one even go about proving such a thing, would I need to look at every person in New York and count each of the hairs on their head? How much time would that take? What if I miscount? What if midway through counting somebody grew hair/got a haircut? Who would actually let some random guy count the hairs on their head? Why does anybody care? All jokes aside, that last question is both the most and least important question on this list, most important for obvious reasons, but least important because it impedes our ability to solve this thought experiment, surprisingly, a majority of math is thought experiments which prepare you for real questions that people care about. Without too much commentary, onto the proof!

It is known(weirdly, that the maximum number of hairs on one’s head is roughly 500,000.) There are currently 8.5 million people in New York. So if everybody had a different number of hairs on their head, there would have to be somebody with 8.5 million hairs on their head, impossible!

Although the answer was simple and the question useless, it is the process and way of thinking outside the box which drew me to math, so despite its simplicity, the process is actually quite deep.


Passion at first I thought of as a hobby, but as I struggled more and more I decided to extend this to include lifestyles. To be clear, I love math. Yet, today I feel as though one gets more relate-ability and humility tied to the mantra “I was always bad at math…” I swear this is the first, most common expression after people learn that I’m a math major. Usually followed by, “what exactly does a mathematician do?” But every time that mantra is uttered, I know that we’ve failed. Not because math didn’t appeal to that person, but because I’m willing to bet that person never even learned what math really is…